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Mirrors > Home > MPE Home > Th. List > funoprab | Structured version Visualization version GIF version |
Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.) |
Ref | Expression |
---|---|
funoprab.1 | ⊢ ∃*𝑧𝜑 |
Ref | Expression |
---|---|
funoprab | ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funoprab.1 | . . 3 ⊢ ∃*𝑧𝜑 | |
2 | 1 | gen2 1797 | . 2 ⊢ ∀𝑥∀𝑦∃*𝑧𝜑 |
3 | funoprabg 7259 | . 2 ⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1535 ∃*wmo 2620 Fun wfun 6335 {coprab 7143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-br 5053 df-opab 5115 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-fun 6343 df-oprab 7146 |
This theorem is referenced by: mpofun 7262 ovidig 7278 ovigg 7281 oprabex 7663 axaddf 10553 axmulf 10554 funtransport 33499 funray 33608 funline 33610 |
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